Genus two KdV soliton gases and their long-time asymptotics

TL;DR

The work develops a rigorous Riemann-Hilbert framework for high-genus KdV soliton gases, focusing on a genus-two gas and its long-time behavior. By solving a genus-two RH problem and employing a two-phase Riemann-Theta structure, it derives precise large-$x$ asymptotics and a five-region long-time classification in the $x$-$t$ plane, including modulated and unmodulated one- and two-phase waves. The analysis relies on careful RH deformations, a scalar g-function, and a genus-two model problem expressed through Theta functions, with extensions to arbitrary genus $\mathcal{N}$ via a corresponding algebro-geometric finite-gap framework. The results connect soliton-gas dynamics to Whitham modulation theory and small-dispersion limits, providing explicit asymptotics that reveal the gas’s structured multi-phase behavior and its impact on the KdV evolution.

Abstract

This paper employs the Riemann-Hilbert problem to provide a comprehensive analysis of the asymptotic behavior of the high-genus Korteweg-de Vries soliton gases. It is demonstrated that the two-genus soliton gas is related to the two-phase Riemann-Theta function as \(x \to +\infty\), and approaches to zero as \(x \to -\infty\). Additionally, the long-time asymptotic behavior of this two-genus soliton gas can be categorized into five distinct regions in the \(x\)-\(t\) plane, which from left to right are rapidly decay, modulated one-phase wave, unmodulated one-phase wave, modulated two-phase wave, and unmodulated two-phase wave. Moreover, an innovative method is introduced to solve the model problem associated with the high-genus Riemann surface, leading to the determination of the leading terms, which is also related with the multi-phase Riemann-Theta function. A general discussion on the case of arbitrary \(N\)-genus soliton gas is also presented.

Figures (18)

• Figure 1: The evolution of the genus two soliton gas potential of the KdV equation at $t = 10$ for parameters $\eta_1 = 0.8$, $\eta_2 = 1.2$, $\eta_3 = 1.6$, $\eta_4 = 2$, and $r_2(\lambda) = 1$. The horizontal axis represents $\frac{x}{4t}$, and the critical points $\eta_1^2$ and $\xi_{\text{crit}}^{(j)}$ for $j = 1, 2, 3$ partition the plane into five distinct regions. These critical values, $\xi_{\text{crit}}^{(j)}$, can be calculated by using equations (\ref{['xi crit']}) and (\ref{['xi critical']}) approximately. For the given parameters, the approximate values are $\xi_{\text{crit}}^{(1)} \approx 2.0905$, $\xi_{\text{crit}}^{(2)} \approx 3.2338$, and $\xi_{\text{crit}}^{(3)} \approx 5.8561$.
• Figure 2: Five asymptotic regions of the genus two KdV soliton gas potential in the $x$-$t$ half plane.
• Figure 3: The jump contour for $Y(\lambda)$ and the associated jump matrices.
• Figure 4: The Riemann surface $\mathcal{S}$ of genus three and its basis of circles.
• Figure 5: The jump contours for $S(\lambda)$ and the associated jump matrices: the gray terms in the matrices vanish exponentially as $x \to +\infty$, and the gray contours also vanish as $x\to +\infty$.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• proof
• Remark 3.1
• Lemma 3.2
• Remark 3.3
• Lemma 3.4
• proof
• Theorem 3.5